A first-order method for solving bilevel convex optimization problems in Banach space

ABSTRACT

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem in Banach space. Regularized forward–backward splitting method is applied to find the minimum like-norm solution of the minimization problem under investigation. We also introduce a gradient-based method, called the minimal like-norm gradient method, for solving this class of problems and establish the convergence of the sequence generated by the algorithm as well as a rate of convergence of the sequence of function values.

KEYWORDS:

• Bilevel optimization problem
• forward–backward splitting method
• minimal like-norm gradient method
• regularization

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 90C25
• 90C30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the Heilongjiang Provincial Natural Sciences Foundation (Grant No. LH2022A017). The work of this author was partially supported by the National Natural Sciences Foundation (Grant No. 11871182).